All Questions
1,570,633
questions
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Epsilon-Delta Theorem for Darboux Integrability
Trying to prove the epsilon delta theorem for Darboux Integrability using the fact that U(f,P)-U(f,Q)≤2NBmesh(P) and L(f,Q)-L(f,P)≤ 2NBmesh(P) where N is the number of elements contained in Q but not ...
0
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0
answers
3
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Proof of the Gelfand-Naimark Theorem
I am reading a proof of the Gelfand-Naimark theorem in Principles of Harmonic Analysis by Anton Deitmar and Siegfried Echterhoff. I have questions about some of the steps.
Theorem. If $A$ is a ...
- 11.4k
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2
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Validity of a proof of the Fisher Information Data Processing inequality, $I[f(X)] \le I[X]$.
I'm trying to prove that taking a function of a random variable never creates a better estimator (in the terms of Fisher information) than using the original random variable directly.
I have a proof (...
- 4,326
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2
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Kronecker sum as a Kronecker product
I seek the following relationship (if there is one so):
𝐶⊗𝐷=(𝐴⊗I)+(I⊗D)
I would like to obtain A=𝑓(C,D)
We can assume D, circulant matrices, square and symmetric for simplicity (then all term of ...
- 19
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3
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Colouring a map with disconnected regions
Consider a map of countries, where each country may consist of up to $n$ disconnected regions. Show that the minimum number of colours needed to colour this map is at most $6n$. We require that all ...
- 45
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2
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Recovery Time for Logistic model equation with harvesting
We are given a logistic growth model with constant harvesting as:
$\frac{dN}{dt} = rN(1-\frac{N}{K})-Y_0$
We are asked to show that the recovery time for harvesting a yield $Y_0$, $T_R(Y_0)$, ...
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answers
5
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Let a,b,c be positive reals such that $a^2 + b^2 + c^2 \leq 3$. Prove $\sqrt {1+a} +\sqrt {1+b} +\sqrt {1+c} \geq \frac{\sqrt 2}{3} *(a+b+c)^2$
I am unsure of where to even start. I know that $$\sqrt 2 \geq \frac{\sqrt 2}{3} *(a^2 +b^2 +c^2)$$ which is needed for solving it, but otherwise I am unsure of what to do.
- 1
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3
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Embedding arbitrary graph in $\mathbb Z^d$ nearest neighbor lattice
Let $G=(V,E)$ be an arbitrary connected graph with infinitely-many vertices and bounded degree $\Delta_G$. I am interested in letting it be a directed graph ($(x,y)\in E$ not necessarily implying $(y,...
- 5,900
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3
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The role of the coefficients in a parabola $\mathcal{P}: y=ax^2+bx+c$
If I have a parabola $\mathcal{P}: y=ax^2+bx+c$, I know that $a$ provides me with the aperture of the parabola; $c$ is the known term. If it is zero the parabola passes through the origin otherwise ...
- 6,411
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5
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Trouble understanding conditions to the Heine definition of the limit
Recall the Heine definition of the limit:
$$
\lim_{x\rightarrow a}f(x)=A \Leftrightarrow (\forall \{x_n\})\left(\lim_{n\rightarrow \infty}f(x_n)=A\right)
$$
, where $\{x_n\}$ is a sequence satisfying:...
- 77
-1
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answers
11
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Prove that $\ Ω(n^n) = \ (3^{2^n})$
Assuming that n is natural,
Is $\ Ω(n^n) = \ (3^{2^n})$ ?
I'm trying to find n0 > 0 and c > 0 so that $\ c*{n^n} <= \ 3^{2^n}$
This is what I've done so far:
$$\ c*{n^n} <= \ 3^{2^n}$$
c = ...
2
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0
answers
12
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Given $f(x+y) = f(x)\cdot f(y)$, prove or disprove $(f(x))^\lambda = f(\lambda x)$ for $\lambda \notin \mathbb{Q}$
I'm messing around with exponentials & logarithms to deepen my familiarity with the basics.
It occurred to me to wonder if $f(x+y) = f(x)\cdot f(y)$ is sufficient to define an exponential, for $f:\...
- 21
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17
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Issues with Lerch Phi and Abel-Plana formula branch cuts & integral divergence
Introduction
I have taken the approach of using the Abel-Plana formula to evaluate the Lerch Phi as follows:
$$
\Phi(z,s,a) = \frac{1}{2 a^s} + \frac{(-\log(z))^{s-1}}{z^a} \Gamma(1-s, -a \log(z)) - 2 ...
- 121
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7
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C projective as C[x] module?
I'm asking myself if $C = C[X]/(x)$ is a projective/flat $C[X]$ module. I found on the internet that this isn't the case, but on the other hand we have $ C[X] = C \oplus xC[X]$, so $C$ is a direct ...
- 91
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13
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Motivating Gauss's suggestions of prize problems for the Goettingen university.
(This question was posted before in History of Science and Mathematics stackexchange, but since I recieved no comments after 6 days, I decided to reask it here.)
P. 220-221 of volume 12 of Gauss's ...
- 912
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7
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Scale box to fill with offset
As shown in the illustration, we have a photo (blue) and a frame (red) that will crop the photo.
The photo has a focal point. We have already translated it such that the focal point aligns with the ...
- 99
2
votes
1
answer
16
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Closedness of operator Hardy type
I am having difficulty proving that the operator
$$ Af(x) = \int_0^x \frac{f(t)}{t}\,dt$$ is not a closed operator from $D(A) = C_c(0,1)$ to $L^2(0,1)$. Is there any easy way to see this?
- 21
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2
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Independence and independent increments
Let $X_1, X_2, X_3$ be three independent identically distributed random variables. Does this mean that
$$ X_3 - X_2, \ X_2 - X_1 $$
are also independent random variables? We can instead inspect the ...
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6
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Determining color of ball from urn with condicion that the two next balls are white
One ball is removed from an urn containing $a$ white and $b$ black balls.
To determine the color of the removed ball, two more balls are drawn. What
is the probability that a black ball has been ...
- 1,650
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8
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Applications of analytic set theory in probability theory
Im studying the theory of analytic sets in polish spaces. In Donald L. Cohn's "Measure Theory" there is mentioned that there are a lot of applications of this in probability theory. But I ...
- 81
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1
answer
18
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Should discrete and continuous time models give the same results?
Today i thought a lot about very simple population models, and there are still a few things that bug me.
Consider a simple discrete exponential growth function:
$$ n(t+1) = n(t) + n(t) b - n(t) d = n (...
- 103
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answers
2
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Trouble with converting the negation of a formula to CNF
I'm trying to convert the negation of the following formula to CNF:
(p → (q → r)) → ((p → q) → (p → r))
These are the steps I am following:
¬((p → (q → r)) → ((p → q) → (p → r)))
¬(¬(¬p ∨ (¬q ∨ r)) ∨ (...
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answers
11
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Efficient way to compute average of outer products.
I have a ser of vectors $v_i \in \mathbb{R}^d$, with $i \in \{1, ..., n\}$. I have to compute a weighted average of the outer products of these vectors:
\begin{equation}
\sum_{i=1}^{n} v_i v_i^T w_i
\...
- 11
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18
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Upper bound of the dot product of two real-valued vectors
I am interested in proving the following:
$ \sum_{i=1}^n a_ib_i \leq \max \{\sum_{i=1}^n a^2_i,\sum_{i=1}^n b^2_i\}$, where $a_i,b_i \in \mathbb{R}$.
I came up with a geometric proof and a proof by ...
- 1
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18
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Property of Lebesgue Density Point
I have this claim concerning a Lebesgue density point that I need to understand rigorously: Let $0$ be a density point of a closed set $A\subset \mathbb{R}^n$. Then for any $x\in\mathbb{R}^n$ there ...
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10
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Question on subspace/dimension
Is it possible to find two vector subspaces $V$ and $W$ of $R^3$ with $V \cap W=\{\vec{0}\}$ so that $\dim V=\dim W =2$?
-2
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Find all solution of $(n-1)\lfloor x \rfloor + n\lceil x \rceil = (n+1)\{x\}$
Given that $n$ is an integer. I have to find all values of $x$ that satisfy the equation.
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6
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Weierstrass-Erdmann condition in light reflection
I have a calculus of variations problem with discontinuities where the Weierstrass-Erdmann condition seems impossible to satisfy. Hoping someone can figure out what I did wrong.
I consider light ...
- 1
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15
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Analogy of Archimedean Property of powers of rational numbers
For any $a,b \in \mathbb{Q}$, $a > 1, b> 0$, can we find an integer $n$ such that $a^n > b$? (Can we prove this without using any property of real numbers and hence functions like logarithm ...
- 309
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1
answer
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Evaluation of a Trigonometric Limit...
$$\lim_{x\to\tfrac{\pi}{4}} \frac{(\cos x + \sin x)^3 - 2\sqrt2}{1 - \sin 2x}$$
I tried solving this question without using L'Hospital's Rule but couldn't find a satisfactory way of approach to solve ...
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0
answers
3
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Evaluate area bounded by curve S=0, its Latus Rectum and X-axis
Several equilateral triangles with respective sides 1, 2 ,3 ...n (n is a natural number) are placed end to end, starting from origin, one side of each lying on X-axis in the first quadrant. If the ...
- 250
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5
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Doubly transitive action on upper half plane.
Can I transform the triangle in the right hand side to a triangle in the left hand side by a Mobius transform?
If $PSL(2, \mathbb{R})$ acts doubly transitive on $\mathbb{R} \cup \infty$, then do we ...
- 139
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7
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Intersection of perpendicular tangent lines - generalization of directrix?
This is a funny little problem that I came up with.
For a differentiable function $f$, define a locus of points $P$ as follows:
Let $m$ be an arbitrary tangent line to $f$, and let $n$ be another ...
- 11
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1
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13
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What are the conditions for a non-Hermitian matric's Rayleigh quotient to be less than its maximum eigenvalue?
What I am trying to prove is that: for a row-stochastic matrix P (not necessarily symmetric), whose every row sums $\sum_{j=1}^nP_{ij}=1$, the Rayleigh quotient of P, $R(\mathbf{x})=\frac{\mathbf{x}^{\...
-3
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answers
39
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$a,b \in \mathbb Z$ and $\frac{ab}{a+b}=2$ How many possible values exist for $a$?
$a,b \in \mathbb Z$ and $\frac{ab}{a+b}=2$
How many possible values exist for $a$?
The question is from my cousin's high school test book. We couldn't figure out how to solve this, any help is ...
- 3
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Expected smallest interval size when intervals are subdivided randomly $N$ times.
Each Iteration: Given a $[0,1]$ interval partitioned into sub-intervals $I_1\ldots,I_n$, choose $k\in[n]$ uniformly at random. Subdivide interval $I_k$ into two disjoint intervals of equal size.
Let $...
- 135
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I'm playing a game where I'm tossing a fair coin. When it's heads I gain a dollar. When it's tails I lose a dollar. I play until I have +24 or -36
I'm playing a game where I'm tossing a fair coin. When it's heads I gain a dollar. When it's tails I lose a dollar. I play until I have +24 or -36. What's the probability of each condition?
I'm ...
kashix9
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1
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38
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What is the generalized solution to $M = \int_0^1 (1-x^m)^n$?
What is the generalized solution to $$M = \int_0^1 (1-x^m)^n \,dx$$
(where it can be expressed in terms of n and m) and m and n are rational numbers if considering m and n as natural numbers makes ...
- 250
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votes
4
answers
37
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If $gcd(a,b)+LCM(a,b)=gcd(a,c)+LCM(a,c)$, then are b and c equal?
Is it true that, if $gcd(a,b)+LCM(a,b)=gcd(a,c)+LCM(a,c)$, then are $b$ and $c$ equal?
For coprime $(a,b)$ and $(a,c)$, it is trivial. It is also easy when $a|b$ and $a|c$. I am unable to construct ...
- 6,718
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19
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Christoffel Symbol in terms of determinant of the metric
It is generally known that, if g is the metric,
$$ Γ^𝜇_{\mu\alpha}= \partial_{\alpha}(ln\sqrt{|g|}).$$
It is also known that Christoffels are symmetric on its second and third components, meaning ...
2
votes
4
answers
79
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How do we evaluate this REALLY tricky integral?
$$\int{\frac{\cos 9x + \cos 6x}{2 \cos 5x-1} dx}$$
The objective is to find the answer in terms of $\sin 4x$ and $\sin x$
I would like to share my attempt and then ask a conceptual doubt as usual but ...
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answers
11
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Showing that $\{x \in X: \sup_n\sum_{i=0}^n f(T^i(x)) = \infty\}$ is a $T$ invariant set
Let $(X, \sigma, \mu)$ be a probability space, $T:X\to X$ a $\mu$-invariant map and $f\in L^1(\mu)$. Define $A := \{x \in X: \sup_n\sum_{i=0}^n f(T^i(x)) = \infty\}$. I would like to conclude that $A$ ...
- 810
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9
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Markov chains: Prove vector of mean hitting times is minimal non-negative solution
Definitions: Let $H^A = \inf \{n \geq 0 : X_n \in A \}$ be the first hitting-time of the set $A$. Let $k_i^A = \mathbb{E}_i(H^A)$ be the mean hitting time of $A$ from $i$.
Theorem: The vector of mean ...
- 81
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14
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Hadamard Gap Theorem and Lacunary Functions
Does anyone know any good reference or even a simple proof for Hadamard Gap Theorem or even just the fact that a lacunary function diverges at 1 (I mean the limit not just the evaluation). In fact, I ...
- 832
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19
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Vector Newton's method without using matrix inverse?
For a set of non liner equations $f_i(\vec x)$ were $i\in \mathbb{N}$ was the index, one can construct a vector
\begin{equation}
F(\vec x)=
\begin{pmatrix}
f_1{\vec x} \\
f_2(\vec x)
\end{pmatrix}
\...
4
votes
3
answers
72
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Evaluate Integral : $\int\;\frac {1}{\sqrt {1-e^{2x}}}dx$
How to evaluate Integral :
$$\int\;\frac {1}{\sqrt {1-e^{2x}}}dx $$
My attemp:
I know that my answer is wrong, but I don't know exactly where the error is. I need help finding out?
$$I = \int\;\frac {...
- 550
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0
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18
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Is the following conjecture relating to the chromatic number correct
Define $\mu(G)$ as the minimum number $k$ so that we can partition $V$ in $n, 1<n<k+1$ sets $V_i$ and so that every set $V_i$ has less than $k$ edges between $V_i$ and $V_i^c$
Let $G=(V,E)$ be ...
- 183
1
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0
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16
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The linear system associated to a blow-up(of surfaces)
Take a point $p\in\mathbb{P}^2_k$ and blow it up. Then we get a morphism $f:X \rightarrow\mathbb{P}^2_k$ from the blow-up. This should then be describe by a linear system on $X$. What is that linear ...
0
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0
answers
13
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Find all subspaces, that are simultaneously invariant relative to linear transformations $A$ and $B$
Find all subspaces, that are simultaneously invariant relative to linear transformations $A$ and $B$, where
$A = \left( \begin{matrix}5 & -1 & -1\\ -1 & 5 & -1 \\ -1 & -1 & 5 \...
- 615
-2
votes
0
answers
24
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Probability of random
In bridge each of the players (A,B,C,D) receives $13$ cards. Suppose A and C have $11$ of $13$ spades between them. What is the probability the remaining two spades are distributed so that B and D ...
- 5